Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,5)(4,8)(6,7)(9,10,13,14)(11,12,15,16)(20,27)(24,28)(26,31)(29,32);(2,7)(3,8)(10,14)(11,15)(17,18)(19,21)(20,28)(22,23)(24,27)(25,30)(26,32)(29,31);(1,3,6,8)(2,4,7,5)(9,11)(10,12)(13,15)(14,16)(18,23)(19,25)(24,28)(26,31);(1,4,6,5)(2,3,7,8)(9,12)(10,11)(13,16)(14,15)(17,19,22,25)(18,21,23,30)(20,26,27,31)(24,29,28,32);(1,2)(3,4)(5,8)(6,7)(17,20,22,27)(18,24,23,28)(19,26,25,31)(21,29,30,32);(1,5,6,4)(2,8,7,3)(17,21)(18,25)(19,23)(20,29)(22,30)(24,31)(26,28)(27,32)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664730);